Introductory Biostatistics

and errors arise in the subtraction of this figure from eachx. This di‰culty can
easily be overcome by using the following shortcut formula for the variance:

s^2 ¼

P

x^2 
ð

P

xÞ^2 =n

n
 1

Our earlier example is reworked in Table 2.10, yielding identical results.

s^2 ¼

ð 548 Þ
ð 56 Þ^2 = 7

6

¼ 16 : 67

When data are presented in the grouped form of a frequency table, the vari-
ance is calculated using the following modified shortcut formula:

s^2 F

P

fm^2 
ð

P

fmÞ^2 =n

n
 1

wherefdenotes an interval frequency,mthe interval midpoint calculated as in
Section 2.2.2 and the summation is across the intervals. This approximation is
illustrated in Table 2.11.

s^2 F

89 ; 724 : 25 
ð 2086 : 5 Þ^2 = 57

56

¼ 238 : 35

sF 15 :4lb

(If individual weights were used, we would haves¼ 15 :9 lb.)
It is often not clear to beginners why we useðn
1 Þinstead ofnas the
denominator fors^2. This number,n
1, called thedegrees of freedom, repre-

TABLE 2.10

xx^2

864

525

416

12 144

15 225

525

749

56 548

NUMERICAL METHODS 79